Numerical approximation of data interpolation least squares method

Claire Chainais-Hillairet

Description

Chapter 1 Numerical approximation of data : interpolation, least squares method

• numerical approximation


• taylor series


• square method


• power functions


• given point


• functions


• polynomial function

Sujets

• Numerical analysis
• Taylor series
• Power function
• Function
• Polynomial

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Chapter 1

Numerical approximation of data : interpolation, least squares method

1

I. Motivation

Approximation of functions

Evaluation of a function

Which functions (f:R→R) can be effectively evaluated in any point ?

Evaluation of a function

Which functions (f:R→R) can be effectively evaluated in any point ?

∈N

➞the power functions :f(x) =x m , m ➞the p 2 x 2 +

functions : a 0 +a 1 x+a

olynomial f(x) =

+a m x m ∙ ∙ ∙

Evaluation of a function

Which functions (f:R→R) can be effectively evaluated in any point ?

➞the power functions :f(x) =x m , m∈N ➞the polynomial functions : f(x) =a 0 +a 1 x+a 2 x 2 +∙ ∙ ∙+a m x m

How can we evaluate other functions in a given point ? for instance :f(x) = cos(x),f(x) = sin(x) exp(x),...

Evaluation of a function

Which functions (f:R→R) can be effectively evaluated in any point ?

➞the power functions :f(x) =x m , m∈N ➞the polynomial functions : f(x) =a 0 +a 1 x+a 2 x 2 +∙ ∙ ∙+a m x m

How can we evaluate other functions in a given point ? for instance :f(x) = cos(x),f(x) = sin(x) exp(x),...

➞approximation by a polynomial function : ★using a Taylor series about the given point, ★a polynomial having the same values as thesearching function in some close points ➡Lagrange interpolation

Principles of Lagrange interpolation

f(x) = sin( πx )(x 2 + 3)

Principles of Lagrange interpolation

f(x) = sin(

πx )(x 2 + 3)

4 points on the curve :

(−1,−4), (1,4), (2,0), (3,−12)

Principles of Lagrange interpolation

f(x) = sin( πx )(x

2 + 3)Lagrange interpolating polynomial

4 points on the curve :

(−1,4), − (1,4), (2,0), (3,−12)

Ppolynomial of degree≤3 satisfying P(−1) =−4 P(1) =4 P(2) =0 P(3) =−12